Hi, I have been pondering about a hypothetical distance between consecutive real numbers. It seems a bit of a paradox, though I expect it will be shown to be a consistent picture. I'll be using recently-learned terminology which hasn't completely set in mind yet, so please have patience with me :) Firstly, let's define a hypothetical function S_r from R->R which in essence is the real analog of the successor function from N->N . I realize that such a function is inherently poorly defined since "succeeding" a real number x by some finite, non-zero number y will give z. But of course, between x and z will be an infinite number of real numbers to fill the interval of y-length. Never-the-less, being a hypothetical question, the terminology should serve for purposes of discussion. Now, let's say we have an x and y in the set of reals satisfying |x-y|=0. From this, it is apparent to me that we can deduce that x=y. However, what I don't understand is why we can't also deduce that x=S_r(y), or that y=S_r(x). What I mean is that if we define y to be the real successor to x, and if the distance between x and y, |x-y|, were some non-zero quantity, then y couldn't be the real successor to x since the interval [x,y] would contain an infinite number of reals n that satisfy x<n<y. So the distance between x and its successor y cannot be greater than 0. In other words, it must equal zero, or |x-y|=0. But from this equation, we were also able to deduce that x=y. Suppose there were a way to rigorously define the distance between consecutive reals. This distance could not be non-zero, so it would have to be based on some definition of zero itself, let's call it null for reference. Now, given a rigorously defined S_r, if x=S_r(y), then |x-y|=null=0. And if |x-y|=0, then x=y. But since x is also equal to S_r(y), then x=y=S_r(y), or in other words, y=S_r(y)? I feel like I'm dressing up something more simple in unnecessary amounts of symbolism. I guess what I'm trying to say is if the distance two consecutive reals can't be non-zero, then it must be zero? And if so, then a real and its real successor must have the same quantity? Thanks for reading!